Earthquake Engineering for Concrete Dams. Anil K. Chopra
Comparison of uniform hazard spectrum and seismic coefficient for concrete dams and buildings.
Source: Adapted from Chopra (1978).
The effective modal earthquake forces may be expressed as the product of the weight of the dam per unit height and a seismic coefficient; its magnitude depends on the pseudo‐acceleration spectral ordinate at the modal period and its height‐wise distribution depends on the shape of the mode. The response of short‐vibration‐period structures, such as concrete gravity dams, is dominated by the fundamental mode of vibration, and the seismic coefficient varies over the dam height, as shown schematically in Figure 1.3.3b. In contrast, traditional analysis and design procedures ignore the dynamic amplification of response, as reflected in the response spectrum and the shape of the mode, and adopt a uniform distribution for the design coefficient (Figure 1.3.3a), resulting in an erroneous distribution of lateral forces and hence of stresses in the dam. The implications of these errors will be discussed in Chapter 7.
To eliminate these errors, it is imperative to consider the dynamics of the system subjected to realistic ground motions in estimating the earthquake response of concrete dams. In Chapters 2–6, such procedures for dynamic analysis of 2D models of gravity dams are developed. In Chapter 7, responses computed by these procedures are demonstrated to be consistent with motions of a gravity dam recorded during an earthquake and with the earthquake performance of Koyna (gravity) Dam.
The traditional design loadings for gravity dams include seismic water pressures in addition to the hydrostatic pressures, as specified by various formulas (U.S. Army Corps of Engineers 1958; Bureau of Reclamation 1966). These formulas differ somewhat in detail and in numerical values but not in underlying assumptions; they are all based on the classical results (Westergaard 1933; Zangar 1952) derived from analyses that assumed the dam to be rigid and water to be incompressible. One of these formulas specifies the seismic water pressure pe = cswH, where c is a coefficient that varies from zero at the water surface to about 0.7 at the reservoir bottom, s is the seismic coefficient, w is the unit weight of water, and H is the total depth of water. For a seismic coefficient of 0.1, the additional water pressure at the base of the dam is about 7% of the hydrostatic pressure; and pressure values at higher elevations are even smaller. As a result, these additional water pressures have little influence on the computed stresses and hence on the geometry of the gravity section that satisfies the traditional design criteria.
Figure 1.3.3 Distribution of seismic coefficients over dam height in traditional design and for the fundamental vibration mode.
Source: Adapted from Chopra (1978).
On the other hand, earthquake‐induced stresses in gravity dams are much larger when dam–water interaction arising from deformations of the dam and water compressibility effects are considered, as will be demonstrated in Chapters 2 and 6. It is apparent, therefore, that hydrodynamic effects are considerably underestimated because of assumptions implicit in traditional design forces.
As mentioned earlier, traditional analysis and design procedures ignore interaction between the dam and foundation. However, such interaction has very significant influence on the dynamics of the system, and, hence, on the earthquake‐induced stresses. This will be demonstrated in Chapters 3 and 6.
Finally, the static overturning and sliding criteria that have been used in traditional design procedures for gravity dams have little meaning in the context of oscillatory response to earthquake motions.
1.4 TRADITIONAL DESIGN PROCEDURES: ARCH DAMS
1.4.1 Traditional Analysis and Design
Traditionally, the dynamic response of the system has not been considered in defining the earthquake forces in the design of arch dams. For example, the U.S. Bureau of Reclamation (1965) stated: “The occurrence of vibratory response of the earthquake, dam, and water is not considered, since it is believed to be a remote possibility.” Thus, the forces associated with the inertia of the dam were expressed as the product of a seismic coefficient – which was constant over the surface of the dam with a typical value of 0.10 or less – and the weight of the dam. Water pressures, in addition to the hydrostatic pressure, were specified in terms of the seismic coefficient and a pressure coefficient that was the same as for gravity dams, defined in Section 1.3.3. This pressure coefficient was based on assumptions of a rigid dam, incompressible water, and a straight dam. Generally, dynamic interaction between the dam and foundation was not considered in evaluating the aforementioned earthquake forces, but in stress analysis of arch dams the flexibility of the foundation sometimes was recognized through the use of Vogt coefficients (Bureau of Reclamation 1965).
The traditional design criteria required that the compressive stress not exceed one‐fourth of the compressive strength or 1000 psi, and the tensile stress should remain below 150 psi.
1.4.2 Limitations of Traditional Procedures
As mentioned in Section 1.3.3 in the context of gravity dams, the seismic coefficient of 0.1 is much smaller than the ordinates of the pseudo‐acceleration response spectra for intense ground motions (Figure 1.3.2). Thus the earthquake forces for arch dams also were greatly underestimated in traditional analysis procedures.
The effective earthquake forces on a dam due to horizontal ground motion may be expressed as the product of a seismic coefficient, which varies over the dam surface, and the weight of the dam per unit surface area. The seismic coefficient associated with earthquake forces in the first two modes of vibration of the dam (fundamental symmetric and anti‐symmetric modes of a symmetric dam) varies, as shown in Figure 1.4.1. In contrast, traditional design procedures ignore the vibration properties of the dam and adopt a uniform distribution for the seismic coefficient, resulting in erroneous distribution of lateral forces and hence of stresses in the dam. A dynamic analysis procedure that eliminates such errors is developed in Chapter 8. Including dam–water–foundation interaction, this procedure is shown in Chapter 10 to produce seismic response results that are consistent with the motions of two arch dams recorded during earthquakes.
As mentioned in Section 1.3.3, the additional water pressures included in traditional design procedures for gravity dams are unrealistically small and have little influence on the computed stresses and hence on the geometry of the dam that satisfies the design criteria. This observation is equally valid for arch dams because the additional water pressures considered for arch dams are similar to those for gravity dams.
Demonstrated in Chapter